Here we use natural units of G=M=c=1, so lengths are in GM/c² and times in GM/c³. The metric signature is time-positive (+,-,-,-). a is the spin parameter (for black holes 0≤a≤M), M the mass equivalent of the total energy of the black hole, and Mirr its irreducible mass:

Shorthand terms:

Covariant metric coeffizients:

Contravariant components (superscripted letters are not powers, but indices):

The dimensionless spin parameter is a=Jc/G/M². Transformation into cartesian coordinates:

Line element in Boyer Lindquist coordinates:

Metric tensor (t,r,θ,Ф):

With the transformation:

where T is a finkelsteinlike time coordinate (radially infalling photons move with dr/dt=1) and ψ the flattened azimuthal angle: